The proof follows closely the arguments in [1]}. First, note that rescaling \(X=X/\Vert X\Vert _\infty \)  does not change the approximating properties of \(\tilde{\beta }_T\)  for the \(\ell _1\) -margin. Indeed, if \(\tilde{\beta }_T\)  fulfills
\(\min _{1 \le i \le n} \frac{Y_i \left\langle \frac{X_i}{\Vert X\Vert _\infty }, \tilde{\beta }_T \right\rangle }{\Vert \tilde{\beta }_T\Vert _1} \ge (1-\varepsilon ) \max _{\beta \ne 0} \min _{1 \le i \le n} \frac{Y_i \left\langle \frac{X_i}{\Vert X\Vert _\infty }, \beta \right\rangle }{\Vert \beta \Vert _1}=:\gamma _R=\gamma /\Vert X\Vert _\infty ,\) 
